Multiple-criteria ranking
using an additive utility function
constructed via ordinal regresion :
UTA method
Roman SłowińskiPoznań University of Technology, Poland
Roman Słowiński
2
Problem statement
Consider a finite set A of alternatives (actions, solutions)
evaluated by n criteria from a consistent family G={g1,...,gn}
g2max
g1(x)
g2(x)
g2min
g1min g1max
A
3
Problem statement
Consider a finite set A of alternatives (actions, solutions)
evaluated by n criteria from a consistent family G={g1,...,gn}
g2(x)
g1(x)
A
4
Problem statement
Taking into account preferences of a Decision Maker (DM),
rank all the alternatives of set A from the best to the worst
A
**
**
x
xx
xx
x
*
*
x
x * *
x x * *
x x x * * x
x x x
x
5
Basic concepts and notation
Xi – domain of criterion gi (Xi is finite or countably infinte)
– evaluation space
x,yX – profiles of alternatives in evaluation space
– weak preference (outranking) relation on X: for each x,yX
xy „x is at least as good as y”
xy [xy and not yx] „x is preferred to y”
x~y [xy and yx] „x is indifferent to y”
n
iiXX
1
6
n
iii aguaU
1
For simplicity: Xi , for all i=1,…,n
For each gi, Xi=[i, i] is the criterion evaluation scale, i i , where
i and i, are the worst and the best (finite) evaluations, resp.
Thus, A is a finite subset of X and
where g(a) is the vector of evaluations of alternative aA on n criteria
Additive value (or utility) function on X: for each aX
where ui are non-decreasing marginal value functions, ui : Xi ,
i=1,...,n
Basic concepts and notation
n
iiiiii ,aa;,A:g
1
g
7
Criteria aggregation model = preference model
To solve a multicriteria decision problem one needs
a criteria aggregation model, i.e. a preference model
Traditional aggregation paradigm:
The criteria aggregation model is first constructed and then applied
on set A to get information about the comprehensive preference
Disaggregation-aggregation (or regression) paradigm:
The comprehensive preference on a subset ARA is known a priori and
a consistent criteria aggregation model is inferred from this information
8
Criteria aggregation model = preference model
The disaggregation-aggregation paradigam has been introduced
to MCDA by Jacquet-Lagreze & Siskos (1982) in the UTA method
– the inferred criteria aggregation
model is the additive value function with piecewise-linear marginal
value functions
The disaggregation-aggregation paradigam is consistent with the
„posterior rationality” principle by March (1988) and the inductive
learning used in artificial intelligence and knowledge discovery
9
The comprehensive preference information is given in form of
a complete preorder on a subset of reference alternatives
ARA,
AR={a1,a2,...,am} – the reference alternatives are rearranged such
that ak ak+1 , k=1,...,m-1
Principle of the UTA method (Jacquet-Lagreze & Siskos, 1982)
A
AR
a1
a2
a5
a6a7
a3a4
10
Example:
Let AR={a1, a2, a3}, G={Gain_1, Gain_2}
Evaluation of reference alternatives on criteria Gain_1, Gain_2:
Reference ranking:
Principle of the UTA method
Gain_1 Gain_2
a1 4 6
a2 5 5
a3 6 4
a1
a2
a3
11
Principle of the UTA method
12
Principle of the UTA method
13
Let’s change the reference ranking:
One linear piece per each marginal value function u1, u2 is not enough
Principle of the UTA method
Gain_1 Gain_2
a1 4 6
a2 5 5
a3 6 4
a1
a3
a2
a1
a2
a3
u1=w1Gain_1, u2=w2Gain_2, U=u1+u2
For a1a3, w2>w1,
but for a3a2, w1>w2,
thus, marginal value functions cannot be linear
14
Principle of the UTA method
15
Principle of the UTA method
The inferred value of each reference alternative aAR:
where
is a calculated value function,
is a value function compatible with
the
reference ranking,
+ and - are potential errors of over- and under-estimation of the
compatible value function, respectively.
The intervals [i, i] are divided into (i–1) equal sub-intervals with
the end points (i=1,...,n)
n
iii ag'ua'U
1
g
aaaUa'U gg
n
iii aguaU
1
g
iiii
iji ,...,j,
jg
11
1
16
Principle of the UTA method
The marginal value of alternative aA is approximated by a linear
interpolation: for 1 ji
jii g,gag
jii
jiij
iji
jiij
iiii gugugg
gagguagu
11
17
Principle of the UTA method
Ordinal regression principle
if then one of the following holds
N.B. In practice, „0” is replaced here by a small positive number that may
influence the result
Monotonicity of preferences
Normalization
11
11
0
0
k~kkk
kkkk
aaa,a
aaa,a
iff
iff
11 kkkk a'Ua'Ua,a gg
n,...,i;,...,jgugu ijii
jii 11101
n,...,iu
u
ii
n
iii
10
11
18
Principle of the UTA method
The marginal value functions (breakpoint variables) are estimated by
solving the LP problem
11
11
0
0
k~kkk
kkkk
Aa
aaa,a
aaa,a
aaFR
iff
iff
to subject
Min
n,...,i;,...,jgugu ijii
jii 11101
ji,Aa,a,a,gu
n,...,iu
u
Rjii
ii
n
iii
and
000
10
11
(C)
19
Principle of the UTA method
If F*=0, then the polyhedron of feasible solutions for ui(gi) is not empty and
there exists at least one value function U[g(a)] compatible with the
complete preorder on AR
If F*>0, then there is no value function U[g(a)] compatible with the
complete preorder on AR – three possible moves:
increasing the number of linear pieces i for ui(gi)
revision of the complete preorder on AR
post optimal search for the best function with respect to Kendall’s in the area F F*+
Jacquet-Lagreze & Siskos (1982)
F F*+
polyhedron of constraints (C)
F= F*
20
Współczynnik Kendalla
Do wyznaczania odległości między preporządkami stosuje się miarę Kendalla
Przyjmijmy, że mamy dwie macierze kwadratowe R i R* o rozmiarze m m, gdzie m = |AR|, czyli m jest liczbą wariantów referencyjnych
macierz R jest związana z porządkiem referencyjnym podanym przez decydenta,
macierz R* jest związana z porządkiem dokonanym przez funkcję użyteczności wyznaczoną z zadania PL (zadania regresji porządkowej)
Każdy element macierzy R, czyli rij (i, j=1,..,m), może przyjmować wartości:
To samo dotyczy elementów macierzy R*
Tak więc w każdej z tych macierzy kodujemy pozycję (w porządku) wariantu a względem wariantu b
gdy ,1
gdy ,50
gdy ,0
ji
i~j
ij
ij
aa
aa.
aalubji
r
21
Współczynnik Kendalla
Następnie oblicza się współczynnik Kendalla :
gdzie dk(R,R*) jest odległością Kendalla między macierzami R i R*:
Stąd -1, 1
Jeżeli = -1, to oznacza to, że porządki zakodowane w macierzach R i R*
są zupełnie odwrotne, np. macierz R koduje porządek a b c d,
a macierz R* porządek d c b a
Jeżeli = 1, to zachodzi całkowita zgodność porządków z obydwu macierzy.
W tej sytuacji błąd estymacji funkcji użyteczności F*=0
W praktyce funkcję użyteczności akceptuje się, gdy 0.75
1
41
mm
*R,Rdk
m
i
m
jijijk *rr*R,Rd
1 121
22
Example of UTA+
Ranking of 6 means of transportation
1
2
23
Preference attitude: „economical”
24
Preference attitude: „hurry”
25
Preference attitude: „hurry”